Download Lab #2 - Duke Physics

Lab #2
Introduction
The goal of the home part of this lab is to measure the impedances of the
capacitor and the inductor and to extract their capacitance and inductance
correspondingly. As part of this project we will convert the output of the Sound
Blaster into a current source by adjusting its output impedance. By analyzing the
ratio of the output impedance of the current source, input impedance of the
oscilloscope, and the impedance of the component under the test, we will find the
regime where both the current source and the oscilloscope work as ideal
instruments and measure the magnitude and the phase of these components.
Home part of the lab
Converting the Sound Blaster output to a current source
An ideal current source has INFINITE output impedance. It drives all current
through the load and there is no current flowing through the current source. The
real current source has a large, but finite, output impedance. It could be represented
by a Norton equivalent circuit (see Figure 1) which is composed of an ideal current
source IN and a parallel resistor RN (see 1.7 of the textbook for reference). This
resistor provides an additional path for current, resulting in smaller current through
the load. This effect is significant when the load impedance is comparable or larger
than the output impedance of the current source.
Fig. 1. Current distribution in a Norton equivalent circuit of a current source with
finite output impedance RN and connected load Rload.
Recalling that Thevenin and Norton circuits describe the same two terminal
networks if their resistors are identical, we can convert the Sound Blaster output to
a current source just by adjusting its Rth output impedance to the desired value (see
Figure 2). The Norton current IN of such a current source depends on VTh and RTh
as:
Fig. 2. Norton equivalent circuit of a current source (a) and its Thevenin
equivalent (b).
Take a 100 KOhm resistor from your resistors kit and convert Channel 1of the
Sound Blaster output to a current source with output impedance of 100 KOhm.
Measuring capacitance of an unknown capacitor
Let’s use the current source we have just built to measure the capacitance of an
unknown capacitor.
Fig. 3. Circuit schematic for measuring the magnitude of the impedance of the
capacitor. Current source is shown as its Thevenin equivalent. Internal input
impedance of the oscilloscope RTh In is shown.
Use the modified Channel 1 of the Sound Blaster to drive current through the
capacitor, and Channel 2 of the Sound Blaster input to measure the voltage V
across the capacitor (see Figure 3). Set the amplitude of VTh to 1 V and the
frequency to 100 Hz. Before taking any measurements we have to analyze the
interplay between the output impedance of the current source RTh Out, the input
impedance of the oscilloscope RTh In (the Sound Blaster input impedance that you
have measured in the previous lab), the impedance of the capacitor ZC() and how
their ratio affects the measurement of the capacitor impedance.
First, the impedance of ZC() and RTh In in parallel (load for the current source)
should be much smaller than the output impedance of the current source RTh Out. In
this case, the current at the output of the current source is equal to IN and it could
be considered as ideal. Second, the capacitor impedance ZC() should be much
smaller than the input impedance of the Sound Blaster RTh In. In this case, almost all
the current IN flows through the capacitor, the measured voltage V depends only on
ZC(), and the Sound Blaster could be considered as an ideal oscilloscope. How do
we make sure that ZC() is small enough to satisfy both conditions if we do not
know its value yet? If you remove the capacitor for a moment, the measured
voltage will be defined by the ratio of RTh Out and RTh In. Now plug the capacitor
back in, and adjust the frequency in such a way that the measured voltage gets
much smaller than it was without the capacitor (keep it high enough to be able
measure it; a few mV amplitude should be enough). Making the voltage with the
capacitor in place much smaller than without the capacitor ensures that ZC() is
much smaller than the input impedance of the Sound Blaster RTh In. If so, than
ZC()||RTh In is almost equal to ZC() and is much smaller than the output
impedance of the current source RTh Out. Measure the amplitude of the voltage
across the capacitor, calculate the magnitude of its complex impedance, and extract
the capacitance (see 2.7 of the text book for reference).
The voltage across the capacitor and the current flowing through it have a
different phase because of the complex nature of a capacitor’s impedance. To
measure this phase shift we have to compare both current and voltage signals. We
are already measuring the voltage V across the capacitor using Channel 2 of the
Sound Blaster. Unfortunately we cannot measure the current through the capacitor
in our current setup. But we do not have to. Instead we can measure VTh, because
its phase is equal to the phase of current IN through the capacitor. Connect the
Channel 1 of the Sound Blaster input directly to the Channel 1 of the Sound Blaster
output to measure VTh.
a)
b)
Fig. 4. (a) Circuit schematic for measuring the magnitude of the impedance and the
phase shift between VTh and voltage across the capacitor (b) and its simplified
version.
Two versions of the complete measurement circuit are shown in Figure 4 a and
b. They are identical, but Figure 4 a shows all the wires as they are (signal and
ground wires go together forming cables, for example), and Figure 4 b is a
simplified version where the wires show just the electrical connections without
showing their real physical positions. It makes the circuit diagram less crowded.
Fig. 5. Screen shot of the Soundcard Oszilloscope program showing the settings
for measuring phase difference between two signals.
Uncheck the Sync check box on the left panel of the Soundcard Oszilloscope
program to adjust independently the voltage scale settings of Channel 1 and
Channel 2 (see Figure 5). Choose Cursors option from Measure drop list and
check Time check box to measure the phase between VTh and VC. Move one
vertical cursor to the point where VTh phase is equal to zero, and another cursor to
the point where voltage across the capacitor has phase equal to zero. dT in the
right bottom corner of the oscilloscope screen shows the time difference between
these two points. Convert this time difference in phase difference. Is measured
phase shift consistent with expected? Report measured capacitance and phase
shift between current flowing through the capacitor and voltage across it.
Show your math.
Measuring inductance of an unknown inductor
Follow the procedure for measuring capacitance and measure the inductance of
the inductor. Report measured inductance and phase shift between current
flowing through the inductor and voltage across it. Show your math.
Note: Take a DMM and measure the resistance of the inductor. Estimate what
would be the voltage cross the inductor if this parasitic resistance is the only
source of the inductor impedance. Keep the frequency of the signal high enough to
have the voltage across the inductor much higher than estimated. In this case the
contribution of this parasitic resistance into the impedance of the inductor will be
negligible and the phase shift between current and voltage signals will correspond
to that of ideal inductor.
In-class part of the lab
The goal of this in-class part of the lab is to study the transient responses of RC,
LC and LCR circuits (see 2.3.1 - 2.3.3 in the text book). You will also learn how to
use the synchronized output of the function generator to trigger the oscilloscope.
We will also start to use Matlab for data processing.
First, you have to build the circuit, shown on Figure. 1. This circuit contains the
inductor L and the capacitor C (components from the home part of this lab), and a
10 KOhm potentiometer R. A potentiometer is a variable resistor with three legs.
Resistance between two of the legs is constant and equal to 10 KOhm. The third
leg is a sliding contact which slides between the first two. By changing its position,
the resistance between the sliding contacts and others could be adjusted to any
value between 0 and 10 KOhm. Use the DMM to identify the legs.
Fig. 6. Circuit schematic for studying the transient response of RC, LC and LCR
circuits. All input, output and parasitic impedances are not shown. Symbols
numbered as 1, 2 and 3 correspond to the 3 BNC connectors on the front panel of
the printed circuit board.
We will use all three BNC connectors available on the print board panel. These
connectors are numbered from 1 to 3 on the circuit diagram (mostly for
convenience). When the function generator is connected to the BNC connector #1,
we have either an LC circuit, if the potentiometer is set to zero resistance, or else
an LCR circuit at any other value of R. When the function generator is connected
to the BNC connector #2, we have an RC circuit. Channel 1 of the oscilloscope is
connected to the BNC connector #3 and measures the voltage across the capacitor.
The circuit diagram shows only physical components. All output (signal generator,
50 Ohm), input (oscilloscope, 1 MOhm) and parasitic (inductor, you have
measured it in the home part of the lab) impedances are not shown. If some of
these impedances affect your next measurements, draw the corresponding
circuit diagram which include those impedances which are important. If you
think that some of these impedances could be neglected, explain why.
One significant difference of our circuit diagram from the text book circuit
diagrams 2.4-2.6 is the presence of the signal generator. We will use it to apply a
square wave signal to the input of the circuit. Every time the input voltage swings
between the high and low values, the voltage across the capacitor gradually decays
from one voltage to another. This transient response depends on the parameters of
the circuit, and is the subject of this part of the lab.
Transient response of the RC circuit
Build the circuit shown in Fig. 1. Set the resistance of the potentiometer to
10 KOhm. Split the signal generator signal using a T adapter, and apply it to the
BNC connector #2 and to Channel 2 of the oscilloscope. Set the wave form of the
signal to a square wave. The amplitude of the signal could be anything; it is just a
matter of convenience. Calculate the time constant =RC of the RC circuit and
report it. Set the period T of the signal so that the voltage across the capacitor has
enough time to completely decay (T>>).
First trigger the oscilloscope with the circuit output signal. Set the trigger slope
to Falling and the trigger level to any value between the minimum and maximum
values of the input signal (page 99 of the User Manual). Because the output voltage
exponentially decays with time, it is hard to set the trigger level so the trigger time
will be fixed in time when the input voltage jumps from one value to another
(actual reference time for the transient response). Play with the trigger level and
you will see what I mean. To do so, we can trigger the oscilloscope with the input
signal. Change the trigger source of the oscilloscope in Channel 2. Now the
reference time is fixed to the moment of the input voltage step.
Save the waveform of the transient response. Login into your lab computer
using your NetID and password. Open Matlab and the script prepared for this lab.
The script is extensively commented and designed to teach you how to import
measured data, plot them, compare with the theoretically predicted exponential
decay, and extract the real time constant of the measured exponential decay. Plot
your data, and save the image for your report. Report the measured time
constant.
Transient response of the LC circuit
Switch the signal generator cable to BNC connector #1. Set the value R of the
potentiometer to zero. According to the circuit diagram, we have now an LC circuit
and the transient response should be a sine wave (see 2.3.2 of the text book). Why
is it not a sine wave but instead it looks like an underdamped LCR circuit?
What is the source of the dissipation?
Transient response of the LCR circuit
Play with the value of the potentiometer resistor R. Watch how the transient
response changed from underdamped to overdamped. Calculate the value of R for
the critically damped LCR circuit. Report it. Show your work.
Take a close look on the input signal. When the LCR circuit is extremely
underdamped the input signal is distorted. Here is a question for an additional 5
points: explain why the input signal is distorted for an underdamped LCR
circuit. Because the input signal is distorted, it is not an ideal source for triggering
the oscilloscope. The function generator has an additional output located on its rear
panel. It is a Sync Out connector. Its output is a 0-5 V square wave synchronized
with the Channel 1 output signal. Set the Sync Output signal (page 2-50 of the
User Manual), connect the Sync Out connector to the Ext Trigger connector of
the oscilloscope, and set the oscilloscope trigger source to External and the trigger
level to 2.5 V. Now you can trigger the oscilloscope with an external signal that is
not connected to the circuit under test (and therefore is not affected by the circuit),
is synchronized with the input signal, and has a conveniently large amplitude.
Save waveforms of underdamped and overdamped waveforms of the LCR
circuit, plot them together and include in your report.